In this section, we reconstruct a linkage from each weak linkage which we have constructed in Section .
We use the the algorithm by Schrijver [1]} using suitable data structures.
Given a weak \(T\) -linkage \(\mathcal {W}\)  of a planar graph, the algorithm by by Schrijver [1]} computes
a \(T\) -linkage of \(G\)  homotopic (and \(F^*\) -homologous) to \(\mathcal {W}\)  in \(O(n^6)\)  time, where \(F^*\)  denotes the innermost face of \(G\) .
In our case, we improve the running time of this algorithm to \(2^{O(k)}n\)  by using suitable data structures and
the properties of the weak linkage \(\mathcal {W}\)  we constructed in Section .
Let \(G^\textsf {p}\)  be the planar graph obtained from \(G^\textsf {rad}\)  by adding \(2^{O(k)}\)  copies of each edge of the frames and skeleton forests
so that the walks of \(\mathcal {W}\)  become pairwise edge-disjoint.
